I need to figure it out how without exceeding the use of the number 10, find three different ways of distributing some blue and red marbles in a container, so that the probability of randomly choosing two marbles of the same color will equal that of different colors. Someone can help?
2026-02-23 08:18:54.1771834734
Fairness problem
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HINT: If you have $r$ red and $b$ blue marbles, there are $\binom{r}2+\binom{b}2$ ways to choose two marbles of the same color, and $br$ ways to choose two marbles of different colors. Use this to write down an equation involving $r$ and $b$, and rearrange it to express $r+b$ in terms of $r-b$. Three small solutions are then quite easy to find by inspection (or a little very easy trial and error).